3. In the previous sections, we learned how to
determine that the two statements
are equivalent. One of these is the equivalent
disjunctive form of a conditional
statement. Here we discuss the different
equivalent foms of conditional
statements.
4. Lessons to be Covered
• Equivalent Forms of the Conditional
• The Inverse, the Converse
and the Contrapositive
5. Equivalent Forms
of the Conditional
A conditional
statement can stated
in many equivalent
forms.Table 5.23 gives
some of the equivalent
statements that can
be used to state a
conditional statement.
6. Example
Write the following in "If
p, then q" statement.
1. A number is divisible by 3,
only if the sum of its digits is
divisible by 3.
2. Every square is a
quadrilateral.
7. Solution
The first statement is of the form
"q, only if p' and the second
is of the form "every p is a q."
1. If the sum of its digits of a number is
divisible by 3, then the number is
divisible by 3.
2. If it is a square, then it is a
quadrilateral.
8. The Converse, the
Inverse, and the
Contrapositive
There are three
statements related to
conditional statements.
These are the
converse, the inverse, and
the contrapositive.
9. Definition
Let p and q are statemnents.
Then the following are the
statements related to p →q.
1. The converse of p→ q is q →p.
2. The inverse of p →q is ~p→~ q.
3. The contrapositive of p →q is q→
p.
10. Example
Write the converse, inverse,
and contrapositive of
If S is a square, then S is a
rectangle
Solution.
1. Converse: If S is a rectangle,
then S
is a square.
2. Inverse: If S is not a square,
then S is a not a rectangle.
3. Contrapositive: If S is not a
rectangle, then S is not a square.
11. All three related statements of the conditional
statement p →q are not
all equivalent to p→q. However, using the truth
table for each related
statements, it can be verified that some of them
are related to one another.
12. These are the following:
Remark 1. Let p and q are statements. Then the
following hold:
13. Example
Determine whether each pair
of statements are equivalent.
1. If you see a man and a woman
holding each others hands, then
are they
are in a relationship with each
other.
If the man and a woman are not
in a relationship with each other,
then
they are not holding each other's
hands.
14.
15. Solution 1. The second statement is the
converse of the first statement.
They are equivalent.
2. The second statement is the
inverse of the first statement.
They are
not equivalent.
